I have made some simple linear interpolation functions:

nM = 7;
xi = Table[2 (j - 1)/(nM - 1), {j, nM}];\[CurlyPhi][1] = 
 Interpolation[{{xi[[1]], 1}, {xi[[2]], 0}, {xi[[nM]], 0}}, 
  InterpolationOrder -> 1]; \[CurlyPhi][2] = 
 Interpolation[{{xi[[1]], 0}, {xi[[2]], 1}, {xi[[3]], 0}, {xi[[nM]], 
    0}}, InterpolationOrder -> 1];
\[CurlyPhi][nM - 1] = 
  Interpolation[{{xi[[1]], 0}, {xi[[nM - 2]], 0}, {xi[[nM - 1]], 
     1}, {xi[[nM]], 0}}, InterpolationOrder -> 1];
\[CurlyPhi][nM] = 
  Interpolation[{{xi[[1]], 0}, {xi[[nM - 1]], 0}, {xi[[nM]], 1}}, 
   InterpolationOrder -> 1];
Do[\[CurlyPhi][j] = 
   Interpolation[{{xi[[1]], 0}, {xi[[j - 1]], 0}, {xi[[j]], 
      1}, {xi[[j + 1]], 0}, {xi[[nM]], 0}}, 
    InterpolationOrder -> 1], {j, 3, nM - 2}];

The Integrate[ ] function can do a definite integral without a problem. For example

\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(2\)]\(\(\[CurlyPhi][
     1]'\)[x] \[DifferentialD]x\)\)

gives the exact answer -1. However the definite integral of the product of two such functions returns unevaluated, even though the piecewise polynomial ought to be integrable. For example

\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(2\)]\(\(\[CurlyPhi][
     1]'\)[x] \(\[CurlyPhi][4]'\)[x] \[DifferentialD]x\)\)

returns

\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(2\)]\(\(\*
TagBox[
RowBox[{"InterpolatingFunction", "[", 
RowBox[{
RowBox[{"{", 
RowBox[{"{", 
RowBox[{"0", ",", "2"}], "}"}], "}"}], ",", "\<\"<>\"\>"}], "]"}],
False,
Editable->False][x]\ \*
TagBox[
RowBox[{"InterpolatingFunction", "[", 
RowBox[{
RowBox[{"{", 
RowBox[{"{", 
RowBox[{"0", ",", "2"}], "}"}], "}"}], ",", "\<\"<>\"\>"}], "]"}],
False,
Editable->False][x]\) \[DifferentialD]x\)\)

What went wrong here?

Thanks in advance for helping!